Rational Curves on Fano Varieties

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Algebraic varieties are common zeros of collections of polynomial equations. Rational curves are the simplest algebraic varieties, and an important approach to study the geometry of algebraic varieties is to study parameter spaces of rational curves contained in them. These parameter spaces are themselves varieties with rich geometry, and their study has broad applications in higher dimensional algebraic geometry, enumerative geometry, arithmetic geometry, and questions inspired by mathematical physics. In this project, several open questions on various aspects of the geometry of spaces of rational curves on algebraic varieties are investigated. The project provides training opportunities for graduate students. The focus of the first part of the project is the study of spaces of rational curves (as well as rational surfaces and linear subvarieties) contained in hypersurfaces. Hypersurfaces of low degree in projective space form an important testing ground for the study of rationally connected and Fano varieties as well as several other questions in birational geometry. Despite some progress over the past few years, some of the basic properties of these spaces are still unknown. A major guiding question for the study of rational curves on hypersurfaces is which Fano hypersurfaces are rational or unirational. The second part of the project is on the study of rational curves on varieties from the perspective of Geometric Maninís conjecture which predicts the growth rate of a counting function associated to the irreducible components of moduli spaces of rational curves on a variety. In this part, several questions on the geometry of spaces of rational curves on Fano threefolds in characteristic zero and on del Pezzo surfaces over fields of finite characteristic are investigated.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.